Characterizations of (4K1, C4, C5)-free graphs

Given a set L of graphs, a graph G is L-free if G does not contain any graph in L as an induced subgraph. There has been keen interest in colouring graphs whose forbidden list L contains graphs with four vertices. A recent paper of Lozin and Malyshev (in press) discusses the state of the art on this problem, identifying three outstanding classes: L=(4K1,claw), L=(4K1,claw, co-diamond), and L=(4K1,C4). In this paper we investigate the class of (4K1, C4, C5)-free graphs and show that if G is a (4K1, C4, C5)-free graph, then G either has bounded clique width or is perfect.